On the existence of geodesic connecting Lagrangian graphs in ℂⁿ

2015/12/28 by Yiyan Xu, Xu, Yiyan
Mathematics · #35-XX #53-XX #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG #msc:35-XX #msc:53-XX

paper · pdf · doi:10.48550/arxiv.1512.08423

19 pages

arxiv created 2015/12/28 · openalex publication_date 2015/12/28 · arxiv updated 2015/12/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper we show that two Lagrangian graphs over the torus in ℂⁿ with large Lagrangian phase can be connected via Lipschitz continuous geodesic with respect to the L² metric on the space of Lagrangian submanifolds. In particular, the geodesic for Lagrangian graphs over the torus in ℂⁿ can be formulated as a degenerate elliptic equation, and we construct geodesic by solving the corresponding Dirichlet problem.

Citations

• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• Calibrated geometries

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